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I'm quiet in a hurry but I think that your question can be deduced as a consequence of the fact that every compact groups has the property (T) of Kazhdan? . In particular finite groups have them. Therefore, your question holds in a much more general context. Does it make sense?

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I'm quiet in a hurry but I think that your question can be deduced as a consequence of the fact that every compact groups has the property (T) of Kazhdan? In particular finite groups have them. Therefore, your question holds in a much more general context. Does it make sense?